Homework 2

Question 1

Some infectious diseases have 2 modes of transmission: via direct contact with an infected individual or indirectly by being exposed to an environmental reservoir (for instance, contaminated water). The diagram on the right shows the transmission of one of these diseases, where infected individuals shed virus into the environment reservoir 𝑉 at a rate 𝜔, and pathogens in this reservoir decay at a rate 𝜌. The per capita birth/death rate of the host is given by 𝜇, and infected individuals recover at rate 𝛾

a.) Write down the system of differential equations (20 points)

b.) Use the Next Generation Method

##Question 2

There is a new outbreak of a respiratory virus on campus. Luckily, there is a vaccine available that provides full protection and immunity doesn’t wane over time. However, Urbana and Champaign decide to proceed with two different strategies to contain this outbreak in the long term. Urbana decides to vaccinate 20% of the newborn individuals (𝑝 = 0.2), while Champaign decides to vaccinate the susceptible population at a rate 𝜈 = 0.2 𝑦𝑟/0. After 200 years, Champaign has eradicated the virus, while in Urbana, the virus became endemic.

a.) Estimate R0.

b.) Estimate the critical proportion of the newborns pc that needs to be vaccinated in order to eradicate the pathogen, and discuss why Urbana did not achieve this goal.

c.) Estimate the critical vaccination rate of susceptible individuals vc necessary to eradicate the virus, and discuss why Champaign achieved this goal.

d.) Plot the dynamics of the infected individuals between years 150 and 200, and discuss how these values are related to the outcome observed in Champaign (disease free) and Urbana (endemicity).

#Champaign

#setup
Csir = function(t, y, pars){
  S = y[1]
  I = y[2]
  R = y[3]
  
  beta <- pars["beta"]
  gamma <- pars["gamma"]
  mu <- pars["mu"]
  v <- pars["v"]
  
  #width(as.list(parmas), {
    dS = mu - beta*S*I - v*S- mu*S
    dI = beta*S*I - gamma*I -mu*I
    dR = gamma*I + v*S- mu*R
    cout <- c(dS, dI, dR)
    list(cout)
  #})

}


#parms

times1 <- seq(0, 200, by=1/365)
paras1 <- c(beta = 265, gamma = 91.25, mu=0.02, v=0.2)
init1 <- c(S=0.999, I=0.001, R=0)


C_out <- ode(y = init1, times = times1, func = Csir, parms = paras1)%>% as.tibble()

## Warning: `as.tibble()` was deprecated in tibble 2.0.0.
## Please use `as_tibble()` instead.
## The signature and semantics have changed, see `?as_tibble`.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_warnings()` to see where this warning was generated.

champlot <- C_out%>% pivot_longer(cols = S:R, 
  names_to = "Variable", values_to = "Proportion") %>%
  ggplot(aes(x = time, y = Proportion, col = Variable)) + 
  geom_line(size = 1) + theme_bw() + xlab("Years")

champlot

#150 years

Champ150 <- C_out %>% filter(time >=150) %>% ggplot(aes(x=time, y=I)) + geom_line(size=1)+ theme_bw()+ labs(x= "Years", y = "Proportion of infected")

Champ150

These graphs show that Champaign has eradicated the infection. We can see that at the 150 year and on there are no oscillations and the proportion is on 0, indicating there are no infections.

Urbana

Usir <- function(t, y, pars){
  S = y[1]
  I = y[2]
  R = y[3]
  
  beta <- pars["beta"]
  gamma <- pars["gamma"]
  mu <- pars["mu"]
  p <- pars["p"]
  
  dS <- mu*(1-p) - beta*S*I - mu*S
  dI <- beta*S*I - gamma*I - mu*I
  dR <- gamma*I + mu*p - mu*R 
  
  Uout <- c(dS, dI, dR)
  list(Uout)
}

times2 <- seq(0, 200, by=1/365)
paras2 <- c(beta = 265, gamma = 91.25, mu=0.02, p=0.2)
init2 <- c(S=0.999, I=0.001, R=0)

U_out <- ode(y = init2, times = times2, func = Usir, parms = paras2)%>% as.tibble()

Uplot <- U_out %>% pivot_longer(cols = S:R, 
  names_to = "Variable", values_to = "Proportion") %>%
  ggplot(aes(x = time, y = Proportion, col = Variable)) + 
  geom_line(size = 1) + theme_bw() + xlab("Years")

Uplot

#150 years

U150 <- U_out %>% filter(time >=150) %>% ggplot(aes(x=time, y=I)) + geom_line(size=1)+ theme_bw()+ labs(x= "Years", y = "Proportion of infected")

U150

Urbana shows oscillations after 150 years, demonstrating that the infection is still on going. As we get to 200 years, we see that the oscillations are decreasing in height. Showing that the efforts to vaccinate are working but at a slower rate to eliminate the infection.